# Pivotal Engineering Journal

### Technical articles from Pivotal engineers.

The random forest has been a burgeoning machine learning technique in the last few years. It is a non-linear tree-based model that often provides accurate results. However, being mostly black box, it is oftentimes hard to interpret and fully understand. In this blog, we will deep dive into the fundamentals of random forests to better grasp them. We start by looking at the decision tree—the building block of the random forest. This work is an extension of the work done by Ando Saabas (https://github.com/andosa/treeinterpreter). Code to create the plots in this blog can be found on my GitHub.

## How Do Decision Trees Work?

Decision trees work by iteratively splitting the data into distinct subsets in a greedy fashion. For regression trees, they are chosen to minimize either the MSE (mean squared error) or the MAE (mean absolute error) within all of the subsets. For classification trees, the splits are chosen so as to minimize entropy or Gini impurity in the resulting subsets.

The resulting classifier separates the feature space into distinct subsets. Prediction of an observation is made based on which subset the observation falls into.

Figure 1: Iterations of a Decision Tree

## Decision Tree Contributions

Let’s use the abalone data set as an example. We will try to predict the number of rings based on variables such as shell weight, length, diameter, etc. We fit a shallow decision tree for illustrative purposes. We achieve this by limiting the maximum depth of the tree to 3 levels.

Figure 2: Decision tree path for predicting number of rings

To predict the number of rings for an abalone, a decision tree will traverse down the tree until it reaches a leaf. Each step splits the current subset into two. For a specific split, the contribution of the variable that determined the split is defined as the change in mean number of rings.

For example, if we take an abalone with a shell weight of 0.02 and a length of 0.220, it will fall in the left-most leaf, with a predicted number of rings as 4.4731. Shell weight will have a contribution of:

``````(7.587 - 9.958) + (5.701 - 7.587) = -4.257
``````

Length will have a contribution of:

``````(4.473 - 5.701) = -1.228
``````

These negative contributions imply that the shell weight and length values for this particular abalone drive its predicted number of rings down.

We can get these contributions by running the below code.

``````from treeinterpreter import treeinterpreter as ti
dt_reg_pred, dt_reg_bias, dt_reg_contrib = ti.predict(dt_reg, X_test)
``````

The variable `dt_reg` is the sklearn classifier object and `X_test` is a Pandas DataFrame or numpy array containing the features we wish to derive the predictions and contributions from. The contributions variable, `dt_reg_contrib`, is a 2d numpy array with dimensions (`n_obs`, `n_features`), where `n_obs` is the number of observations and `n_features` is the number of features.

We can plot these contributions for a given abalone to see which features most impact its predicted value. We can see from the below plot that this specific abalone’s weight and length values negatively impact its predicted number of rings.

Figure 3: Contribution plot for one example (Decision Tree)

We can compare this particular abalone’s contributions to the entire population by using violin plots. This overlays a kernel density estimate onto the plot. In the below figure, we see that the abalone’s shell weight is abnormally low compared to the rest of the population. In fact, many of abalones have a shell weight value which gives it a positive contribution.

Figure 4: Contribution plot with violin for one observation (Decision Tree)

The above plots, while insightful, still do not give us a full understanding on how a specific variable affects the number of rings an abalone has. Instead, we can plot a given feature’s contribution against its values. If we plot the value of shell weight compared to its contribution, we gain the insight that increasing shell weight results an increase in contribution.

Figure 5: Contribution vs. shell weight (Decision Tree)

Shucked weight, on the other hand, has a non-linear, non-monotonic relation with the contribution. Lower shucked weight values have no contribution, higher shucked weight values have negative contribution, and in between, the contribution is positive.

Figure 6: Contribution vs. shucked weight (Decision Tree)

## Extending to Random Forests

This process of determining the contributions of features can naturally be extended to random forests by taking the mean contribution for a variable across all trees in the forest.

Figure 7: Contribution plot with violin for one observation (Random Forest)

Because random forests are inherently random, there is variability in contribution at a given shell weight. However, the increasing trend still remains as shown by the smoothed black trend line. As with the decision tree, we see that increasing shell weight corresponds to an higher contribution.

Figure 8: Contribution vs. shell weight (Random Forest)

Again, we may see complicated, non-monotonic trends. Diameter appears to have a dip in contribution at about 0.45 and a peak in contribution around 0.3 and 0.6. Apart from that, there seems to be a general increasing relationship between diameter and number of rings.

Figure 9: Contribution vs. diameter (Random Forest)

## Classification

We have shown that feature contribution for regression trees is derived from the mean number of rings and how it changes at successive splits. We can extend this to binomial and multinomial classification by looking instead at the percentage of observations of a certain class within each subset. The contribution for a feature is the total change in the percentage caused from that feature.

This is more easily explained with an example. Suppose we instead are trying to predict sex, i.e., whether the abalone is a female, male, or an infant.

Figure 10: Decision Tree path for multinomial classification

Each node has 3 values—the percentage of abalones in the subset that are female, male, and infants respectively. An abalone with a viscera weight of 0.1 and a shell weight of 0.1 would end up in the left-most leaf (with probabilities of 0.082, 0.171, and 0.747). The same logic of contributions for regression trees applies here.

The contribution of viscera weight to this particular abalone being in infant is:

``````(0.59 - 0.315) = 0.275
``````

And the contribution of shell weight is:

``````(0.747 - 0.59) = 0.157
``````

We can plot a contribution plot for each class. Below, we have shown one such plot for the infant class.

Figure 11: Contribution plot with violin for one infant observation (Multi-Class Decision Tree)

And as before, we can also plot the contributions vs. the features for each class. The contributions for the abalone being female increases as shell weight increases while the contribution for it being an infant decreases. For males, the contribution increases initially and then decreases when shell weight is above 0.5.

Figure 12: Contribution vs. shell weight for each class (Random Forest)

## Final Thoughts

We have shown in this blog that by looking at the paths, we can gain a deeper understanding of decision trees and random forests. This is especially useful since random forests are an embarrassingly parallel, typically high performing machine learning model. To satisfy Pivotal’s clients’ business needs, we not only need to deliver a highly predictive model, but a model that is also explainable. That is, we do not want to give them a black box, regardless of how well it performs. This requirement is important when dealing with clients in government or the financial space since our models will need to be passed through compliance.